## DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 351-356

## 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

### DMTCS Conference Volume AE (2005), pp. 351-356

author: Jun Tarui On the Minimum Number of Completely 3-Scrambling Permutations A family P = {π 1 ,…,π q } of permutations of [n]={1,…,n} is completely k -scrambling [Spencer, 1972; Füredi, 1996] if for any distinct k points x 1 ,…,x k ∈[n] , permutations π i 's in P produce all k! possible orders on π i (x 1 ),…,π i (x k ) . Let N * (n,k) be the minimum size of such a family. This paper focuses on the case k=3 . By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. 2 /  log 2 e log 2 n ≤ N * (n,3) ≤2 log 2 n + (1+o(1)) log 2 log 2 n. We also prove the existence of lim n→∞ N * (n,3) / log 2 n = c 3 . Determining the value c 3 and proving the existence of lim n→∞ N * (n,k) / log 2 n = c k for k ≥4 remain open. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. Jun Tarui (2005), On the Minimum Number of Completely 3-Scrambling Permutations, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 351-356 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAE0168.ps.gz (65 K) dmAE0168.ps (153 K) dmAE0168.pdf (151 K)

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